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Theorem cdeqbox 25132
Description: Distribute conditional equality over 'always'. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqbox.1  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cdeqbox  |- CondEq ( x  =  y  ->  ( [.] ph  <->  [.] ps ) )

Proof of Theorem cdeqbox
StepHypRef Expression
1 ax-lll 25130 . . 3  |-  ( x  =  y  ->  [.] x  =  y )
2 alneal1 25103 . . . . 5  |-  ( [.] x  =  y  ->  x  =  y )
3 cdeqbox.1 . . . . . 6  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
43cdeqri 2990 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
52, 4syl 15 . . . 4  |-  ( [.] x  =  y  -> 
( ph  <->  ps ) )
65boxbid 25114 . . 3  |-  ( [.] x  =  y  -> 
( [.] ph  <->  [.] ps )
)
71, 6syl 15 . 2  |-  ( x  =  y  ->  ( [.] ph  <->  [.] ps ) )
87cdeqi 2989 1  |- CondEq ( x  =  y  ->  ( [.] ph  <->  [.] ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632  CondEqwcdeq 2987   [.]wbox 25073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl1 25077  ax-ltl2 25078  ax-ltl3 25079  ax-ltl4 25080  ax-lmp 25081  ax-nmp 25082  ax-ltl5 25096  ax-ltl6 25097  ax-lll 25130
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-cdeq 2988  df-dia 25083
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